Put the roast in a large aluminum plate (for barbecue) or in a roasting pan (for oven).
Combine in a bowl, garlic, ginger, brown sugar, hot pepper, thym, allspice, salt, and cinnamon.
Add 1 tablespoon of coconut milk to the mixture.
Brush the roast with the mixture and let it rest for 30 minutes at room temperature.

Meanwhile, preheat the barbecue to maximum intensity or the oven to 350 F (180 C).
Mix the remaining coconut milk and lime juice, and pour the sauce into the plate, around the roast.
Cover the plate with aluminum foil, making sure to seal it tightly but without touching the roast.

When using the barbecue method, turn off one side of the barbecue and reduce the other elements to medium-high heat or just enough to maintain a temperature of 350 F (180 C).
Place the plate on the unlit side of the barbecue or put it in the oven.
Grill or roast for 2 hours and 30 minutes.

Uncover and baste with the cooking juices.
Continue cooking, uncovered, for an additional 30 to 45 minutes, basting twice, until the meat is tender with a fork or a meat thermometer indicates 170 F (75 C).
Cover loosely with aluminum foil and let it rest for 15 minutes before slicing.
Transfer the roast to a cutting board and discard the cooking juices.
Cut the strings and slice the roast.

Spicy potatoes in foil:
Preheat the barbecue to medium-high heat or the oven to 350 F (180 C).

Brush the potatoes under running water. Pat them dry and cut them into 8 pieces lengthwise.
Place them on a large sheet of heavy aluminum foil.

Mash in a bowl, the butter, hot pepper powder, salt, and pepper.
Sprinkle the potatoes with the spicy butter mixture.
Cover with a large sheet of aluminum foil and seal the edges hermetically, folding them.
Slide the foil package onto a tray.
Grill or bake for 30 minutes, taking care to gently shake the package a few times and flip it halfway through cooking.
Open a small section carefully and check with a fork if the potatoes are tender.
If they are not, return them to cook for an additional 5 to 10 minutes.

Prepare the dipping sauce by combining all the ingredients in a bowl.

Let the potatoes rest for 5 minutes.
Open the foil package and remove the potatoes with a spatula (it may be necessary to scrape off the ones that are particularly crispy).
Serve with the dipping sauce.